First-year courses · Mathematics

Linear algebra I: systems, matrices and vectors

Row reduction, matrix algebra, determinants, dot and cross products, span, independence, basis and eigenvalues: first-year linear algebra flashcards.

45 cards6 topicsAbout 20 minutes a roundTypical first-year introductory linear algebra (matrices, vectors, systems of equations)

Print cards to cut out (PDF)Print a study list (PDF)

About this deck

A first course in linear algebra moves quickly from solving systems of equations to abstract ideas about vector spaces, and this deck follows the same path. It covers augmented matrices and row reduction, echelon forms, pivots and free variables, matrix multiplication and inverses, determinants and their properties, vectors in R^2 and R^3 with dot and cross products, projections and planes, then span, linear independence, basis, dimension, subspaces, linear transformations and an introduction to eigenvalues and eigenvectors.

High school algebra is enough to begin; you do not need calculus for most of this deck. It does help to be careful with arithmetic, because one sign error in row reduction ruins everything that follows.

The definitions in linear algebra are precise, and instructors test them word for word, so for the Vector spaces tag try reciting each definition in full before flipping, then give your own example and a non-example. For computational cards, redo the example by hand rather than just reading it. Mixing tags is especially useful here: the Invertible matrix theorem card ties together determinants, row reduction, independence and span, and noticing those links is what makes later topics easy.

Common mistakes: assuming AB = BA (matrix multiplication is not commutative); thinking det(A + B) = det A + det B (false, although det(AB) = det A det B is true); and calling a set a subspace without checking that it contains the zero vector. Students also often confuse the span of a set (all combinations) with the set itself.

Before you start: High school algebra; comfort solving two or three linear equations by elimination.

All 45 cards

Cover the definitions with a sheet of paper and test yourself, or use the study modes above.

Systems and row reduction

Linear system: A set of equations in which each unknown appears only to the first power with constant coefficients. It has no solution, exactly one, or infinitely many.Example: x + 2y = 5 and 3x - y = 1 meet at the single point (1, 2).
Augmented matrix: The coefficients of a linear system written as rows, with the constants added as a final column after a vertical bar.Example: x + 2y = 5, 3x - y = 1 becomes [1 2 | 5; 3 -1 | 1].
Elementary row operations: Swap two rows; multiply a row by a nonzero constant; add a multiple of one row to another. None of them changes the solution set.Example: R2 - 3R1 turns [1 2 | 5; 3 -1 | 1] into [1 2 | 5; 0 -7 | -14].
Row echelon form: Zero rows are at the bottom, each leading nonzero entry is to the right of the one above it, and all entries below a leading entry are 0.Example: [1 2 3; 0 4 5; 0 0 6] has this staircase shape.
Reduced row echelon form (RREF): An echelon form where every leading entry is 1 and is the only nonzero entry in its column. Each matrix has exactly one such form.Example: [1 0 2; 0 1 -1; 0 0 0].
Gaussian elimination: Using row operations to reach echelon form, then solving by back substitution (or continuing to reduced form, called Gauss-Jordan elimination).Example: From [1 2 | 5; 0 -7 | -14]: y = 2, then x = 5 - 4 = 1.
Pivot: The position of a leading entry in an echelon form of a matrix; columns holding one correspond to basic (non-free) variables.Example: In [1 2 3; 0 0 4], the pivots are in columns 1 and 3.
Free variable: A variable whose column has no pivot; it can take any value, and the basic variables are written in terms of it.Example: x + y + z = 1: y = s and z = t are free, so x = 1 - s - t.
Inconsistent system: A system with no solution; row reduction produces a row of the form [0 0 ... 0 | c] with c not 0.Example: x + y = 1 and x + y = 3 reduce to [0 0 | 2]: no solution.
Homogeneous system: A system Ax = 0. It always has the trivial solution x = 0, and has nontrivial solutions exactly when there is a free variable.Example: Two equations in three unknowns, all equal to 0, always have infinitely many solutions.
Rank and the rank-nullity theorem: Rank is the number of pivots of A. For an m by n matrix, rank plus nullity (dimension of the null space) equals n, the number of columns.Example: A 3 by 5 matrix of rank 3 has a null space of dimension 2.

Matrix algebra

Matrix multiplication: Entry (i, j) of AB is the dot product of row i of A with column j of B; AB is defined only when A has as many columns as B has rows.Example: [1 2; 3 4] times the column [5; 6] gives [17; 39].
Non-commutativity of matrix products: In general AB and BA are different, and one may be defined while the other is not, so the order of factors matters.Example: A = [0 1; 0 0], B = [0 0; 1 0]: AB = [1 0; 0 0] but BA = [0 0; 0 1].
Transpose: The matrix A^T obtained by turning rows into columns. Key rule: (AB)^T = B^T A^T.Example: [1 2 3; 4 5 6]^T = [1 4; 2 5; 3 6].
Identity matrix: The square matrix I with 1s on the main diagonal and 0s elsewhere; AI = IA = A for any square A of the same size.Example: The 2 by 2 version is [1 0; 0 1].
Inverse of a matrix: For square A, the matrix A^(-1) with A A^(-1) = A^(-1) A = I. For [a b; c d] it is (1/(ad - bc))[d -b; -c a], when ad - bc is not 0.Example: [2 1; 5 3] has ad - bc = 1, so its inverse is [3 -1; -5 2].
Finding an inverse by row reduction: Row reduce the block matrix [A | I]. If the left side becomes I, the right side is A^(-1); if a zero row appears on the left, A is not invertible.Example: [1 2; 3 4] gives [1 0 | -2 1; 0 1 | 1.5 -0.5].
Invertible matrix theorem: For square A these are equivalent: A is invertible; det A is not 0; RREF of A is I; Ax = b has a unique solution for every b; columns are independent and span R^n; 0 is not an eigenvalue.Example: [1 2; 2 4] has determinant 0, so all of these fail together.
Symmetric matrix: A square matrix equal to its own transpose, so the entry in row i, column j equals the entry in row j, column i.Example: [2 7; 7 5] satisfies A^T = A.

Determinants

Determinant of a 2 by 2 matrix: For [a b; c d], the number ad - bc. It is nonzero exactly when the matrix is invertible.Example: [3 2; 1 4]: 12 - 2 = 10.
Cofactor expansion: Computing a determinant by expanding along a row or column: sum of each entry times (-1)^(i+j) times the determinant of the matrix with that row and column removed.Example: [1 2 3; 0 4 5; 1 0 6]: 1(24) - 2(0 - 5) + 3(0 - 4) = 22.
Row operations and determinants: Swapping two rows changes the sign; multiplying a row by k multiplies the determinant by k; adding a multiple of one row to another changes nothing.Example: If det A = 5, swapping two rows and then doubling one row gives -10.
Product and transpose rules: det(AB) = det(A) det(B); det(A^T) = det(A); det(A^(-1)) = 1/det(A). But the determinant of a sum is generally not the sum of determinants.Example: If det A = 2 and det B = 3, then det(AB) = 6 and det(A^(-1)) = 1/2.
Determinant of a triangular matrix: For an upper or lower triangular matrix it is the product of the diagonal entries.Example: [2 5 1; 0 3 7; 0 0 4] has determinant 2 * 3 * 4 = 24.
Geometric meaning of a determinant: Its absolute value is the factor by which the matrix scales areas (in 2D) or volumes (in 3D); a negative sign means orientation is flipped.Example: [2 0; 0 3] maps the unit square to a 2 by 3 rectangle of area 6.
Cramer's rule: For an invertible system Ax = b, each unknown x_i = det(A_i)/det(A), where A_i is A with column i replaced by b.Example: x + 2y = 5, 3x - y = 1: det A = -7, det A_1 = [5 2; 1 -1] = -7, so x = 1.

Vectors and geometry

Norm (length) of a vector: The square root of the sum of the squares of the components; it measures the vector's length.Example: (3, 4) has length sqrt(9 + 16) = 5.
Dot product: u . v = sum of products of matching components; it also equals |u||v| cos(theta), where theta is the angle between them.Example: (1, 2, 3) . (4, -5, 6) = 4 - 10 + 18 = 12.
Orthogonal vectors: Two vectors whose dot product is 0; geometrically they meet at a right angle.Example: (2, -1) . (1, 2) = 2 - 2 = 0.
Projection of u onto v: The component of u in the direction of v: (u . v / v . v) v.Example: u = (2, 3), v = (1, 0): projection (2, 0).
Cross product: For u, v in R^3, the vector (u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1). It is perpendicular to both, and its length is the area of their parallelogram.Example: (1, 2, 3) x (4, 5, 6) = (-3, 6, -3).
Equation of a plane: With normal vector n = (a, b, c) through point P0: a(x - x0) + b(y - y0) + c(z - z0) = 0.Example: Normal (1, 2, 3) through (1, 0, 0): x + 2y + 3z = 1.
Vector equation of a line: x = p + t d: all points reached from a point p by moving any multiple t of a direction vector d.Example: Through (1, 0, 2) with direction (3, -1, 1): (1 + 3t, -t, 2 + t).

Vector spaces

Linear combination: A sum of scalar multiples of vectors: c1 v1 + c2 v2 + ... + ck vk.Example: 2(1, 0) + 3(0, 1) = (2, 3).
Span: The set of all linear combinations of a given list of vectors.Example: The span of (1, 0) and (0, 1) is all of R^2; the span of (1, 2) alone is a line through the origin.
Linear independence: Vectors are independent when c1 v1 + ... + ck vk = 0 forces every c_i = 0; otherwise one of them is a combination of the others.Example: (1, 2) and (2, 4) are dependent because 2(1, 2) - (2, 4) = 0.
Basis: A set of vectors that is linearly independent and spans the space, so every vector in the space has exactly one representation using it.Example: (1, 0, 0), (0, 1, 0), (0, 0, 1) is the standard one for R^3.
Dimension: The number of vectors in any basis of a space; all bases of the same space have the same count.Example: A plane through the origin in R^3 has dimension 2.
Subspace: A subset that contains the zero vector and is closed under addition and scalar multiplication.Example: The line y = 2x in R^2 qualifies; y = 2x + 1 does not, since it misses (0, 0).
Column space and null space: Col(A) is the span of the columns of A (all possible outputs Ax); Nul(A) is the set of all solutions to Ax = 0.Example: For [1 2; 2 4], Col(A) is the line spanned by (1, 2) and Nul(A) is spanned by (-2, 1).
Linear transformation: A function T between vector spaces with T(u + v) = T(u) + T(v) and T(cu) = cT(u). On R^n it is always multiplication by some matrix.Example: Rotation by 90 degrees counterclockwise in R^2 has matrix [0 -1; 1 0].

Eigenvalues

Eigenvalue and eigenvector: A nonzero vector v with Av = lambda v for some scalar lambda: A only stretches or flips v. The scalar lambda is the eigenvalue.Example: A = [4 1; 2 3], v = (1, 1): Av = (5, 5) = 5v, so lambda = 5.
Characteristic equation: det(A - lambda I) = 0; its roots are the eigenvalues of A.Example: [4 1; 2 3]: (4 - lambda)(3 - lambda) - 2 = lambda^2 - 7 lambda + 10 = 0, so lambda = 2 or 5.
Trace and determinant checks: The eigenvalues (with multiplicity) add up to the trace (sum of diagonal entries) and multiply to the determinant.Example: [4 1; 2 3]: 2 + 5 = 7 = 4 + 3 and 2 * 5 = 10 = 12 - 2.
Diagonalization: Writing A = P D P^(-1) with D diagonal; possible exactly when an n by n matrix has n linearly independent eigenvectors (columns of P, eigenvalues on D).Example: Then A^k = P D^k P^(-1), which makes high powers easy to compute.